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ISI2021-MMA: 19

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Suppose $G$ is a cyclic group and $a, b \in G$. There does not exist any $x \in G$ such that $x^{2}=a$. Also, there does not exist an $y \in G$ such that $y^{2}=b$. Then,

  1. there exists an element $g \in G$ such that $g^{2}=a b$.
  2. there exists an element $g \in G$ such that $g^{3}=a b$.
  3. the smallest exponent $k>1$ such that $g^{k}=a b$ for some $g \in G$ is $4 .$
  4. none of the above is true.
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